HW2_Sol - Math 104 Fall 2009 Homework 2: solution set We...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 104 Fall 2009 Homework 2: solution set We use the notations below to ease readability. Matrices are bold capital, vectors are bold lowercase and scalars or entries are not bold. For instance, A is a matrix and a ij (sometimes, A ( i,j )) its ( i,j )th entry. Likewise x is a vector and x j its j th component. The linear span generated by a group of vecotors v 1 , v 2 ,..., v n is denoted by span( v 1 , v 2 ,..., v n ). Problem 1 Since ( AB )( B - 1 A - 1 ) = A ( BB - 1 ) A - 1 = AIA - 1 = AA - 1 = I , we have that AB is invertible and the inverse of AB is B - 1 A - 1 . Problem 2 Suppose that R = r 11 ··· r 1 m . . . . . . . . . r m 1 ··· r mm = [ r 1 , ··· , r m ] is a nonsingular upper-triangular matrix, which means { r 1 , ··· , r m } forms a basis of C m , and r ij = 0 for i > j . Suppose e j is the canonical unit vector with 1 in the j th entry and zeros elsewhere. Then r j = j i =1 r ij e i , which implies r j span( e 1 ,..., e j ) and so span(
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 12/05/2010.

Page1 / 2

HW2_Sol - Math 104 Fall 2009 Homework 2: solution set We...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online